Reception and reconstruction of analog signals are performed in a wide variety of applications, including wireless communication systems, spectrum management applications, radar systems, medical imaging systems and many others. In many of these applications, an information-carrying analog signal is sampled, i.e., converted into digital samples. The information is then reconstructed by processing the digital samples.
The minimum sampling rate needed for perfect reconstruction of an arbitrary sampling method is known. Further, various methods for signal sampling and reconstruction are known in the art. Some sampling and reconstruction methods refer to bandpass or band-limited signals, and in particular to multi-band signals, i.e., signals that are confined to a finite set of spectral bands. Additionally known are methods for periodic non-uniform sampling of multi-band signals and methods for sampling and reconstruction of multi-band signals.
Various applications, particularly in the field of secure communications, require the production of truly random numbers at a high bit-rate. Most current random number generators (RNGs) typically employ complicated, yet ultimately deterministic, calculations, generating numbers that are, at best, pseudo-random. Other methods employ the inherent, and essential, randomness of quantum processes, since, in accordance with the laws of physics, there is no way, even in theory, to find a pattern within random numbers generated from quantum measurement. Such methods include radioactive decay (see, for example, U.S. Pat. No. 6,445,217, to Figotin, et al., issued Jun. 1, 2004) or outputs of a beam splitter to establish random numbers from the path of a photon (U.S. Pat. No. 6,309,139, to Dultz et al., issued Aug. 19, 2003). A further method uses thermodynamic processes such as diode current fluctuations or Johnson noise measured on the voltage across a resistor (see, for example, U.S. Pat. No. 6,271,263, to Nagai, issued May 27, 2003).
Compressive sensing is a promising new field that has unlocked novel devices such as the single pixel camera. Many demonstrations of compressive sensing involve a high speed clock somewhere in the signal chain, diminishing the advantages of slow and brief sampling that compressive sensing offers.
Compressive sensing is a technique for finding sparse solutions to underdetermined linear systems. An underdetermined system of linear equations has more unknowns than equations and generally has an infinite number of solutions. However, if there is a unique sparse solution to the underdetermined system, then the compressed Sensing framework allows the recovery of that solution. In electrical engineering, particularly in signal processing, compressed sensing is the process of acquiring and reconstructing a signal that is supposed to be sparse or compressible.
Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The Nyquist theorem states:                If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.        
In essence, the theorem shows that a band-limited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal
By extending the all-or-nothing conditional of the Nyquist theorem, compressive sensing promises to open up entire new realms of sensing techniques. In the areas of communications and RF signal detection, compressive sensing can potentially enable the capture and reconstruction of signals over large bandwidth with small, low-power hardware and extremely small numbers of samples.
Initial demonstrations of compressive RF sampling consisted of capturing high-bandwidth sparse signals with fast digitizers, which handily satisfy the Nyquist condition, followed by post-selecting a random subset of samples. Hypothetically, were one to only have access to that limited, random subset, one could successfully reconstruct the signal with high probability. Of course, the initial investment in performing high-speed digital sampling of the signal negates the need for the compressive alternative.
A second generation of compressive RF sensors developed around the concept of random demodulation in which the signal of interest is mixed with a pseudo-random phase sequence and sampled with a lower-speed ADC. While this approach provides power savings with its lack of a high-speed ADC, it still requires high-speed digital hardware to generate the pseudo-random sequences. In this way, it does not fully realize the advantages of compressive sensing for RF signals.
What is needed are improved systems and methods for performing compressive sensing.